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Sundman of Helsingfors, utilizing analytical methods due to the
Italian Levi-Civita and the French Painleve, and making an
ingenious transformation of his own, proved the existence of a
solution in the sense described above. Sundman's solution is
not adapted to numerical computation, nor does it give much
information regarding the actual motion, hut that is not the
point of interest here: a problem which had not been known to
be solvable was proved to be so. Many had struggled desperately
to prove this much; when the proof was forthcoming, some,
humanly enough, hastened to point out that Sundman had
done nothing much because he had not solved some problem
other than the one he had. This kind of criticism is as common
in mathematics as it is in literature and art, showing once more
that mathematicians are as human as anybody*
Poincare's most original work in mathematical astronomy
was summed up in his great treatise Les mlthodes nouvettes de fa
me'canique celeste (New Methods of Celestial Mechanics; three
volumes, 1892,1893,1899). This was followed by another three-
volume work in 1905-10 of a more immediately practical
nature, Lemons de mecanique c€lesle, and a little later by the
publication of his course of lectures Sur le$ figures d'e'quilibre
tfunemassejluide (On the Figures ofEquilibrium of aFluid Mass),
and a historical-critical book Sur les hypotheses cosmogoniqwea
(On Cosmological Hypotheses),
Of the first of these works Darboux (seconded by many
others) declares that it did indeed start a new era in celestial
mechanics and that it is comparable to the Mecanique celeste ot
Laplace and the earlier work of D'Alembert on the precession
of the equinoxes. 'Following the road in analytical mechanics
opened up by Lagrange,* Darboux says, *... Jacobi had estab-
lished a theory which appeared to be one of the most complete
in dynamics. For fifty years we lived on the theorems of the
illustrious German mathematician, applying them and studying
them from all angles, but without adding anything essential
It was Poincare1 who first shattered these rigid frames in whicih
the theory seemed to be encased and contrived for it vistas and
new windows on the external world. He introduced or used, in
the study of dynamical problems, different notions: the first,